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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Semilattices of finitely generated ideals of exchange rings with finite stable rank
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by F. Wehrung PDF
Trans. Amer. Math. Soc. 356 (2004), 1957-1970 Request permission

Abstract:

We find a distributive $(\vee ,0,1)$-semilattice $S_{\omega _1}$ of size $\aleph _1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular:

  • [—] There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to $S_{\omega _1}$.

  • [—] There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to $S_{\omega _1}$.

  • These results are established by constructing an infinitary statement, denoted here by $\mathrm {URP_{sr}}$, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice $S_{\omega _1}$.

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    Additional Information
    • F. Wehrung
    • Affiliation: Département de Mathématiques, CNRS, UMR 6139, Université de Caen, Campus II, B.P. 5186, 14032 Caen Cedex, France
    • MR Author ID: 242737
    • Email: wehrung@math.unicaen.fr
    • Received by editor(s): January 3, 2003
    • Received by editor(s) in revised form: April 2, 2003
    • Published electronically: October 28, 2003
    • © Copyright 2003 American Mathematical Society
    • Journal: Trans. Amer. Math. Soc. 356 (2004), 1957-1970
    • MSC (2000): Primary 06A12, 20M14, 06B10; Secondary 19K14
    • DOI: https://doi.org/10.1090/S0002-9947-03-03369-5
    • MathSciNet review: 2031048